Evaluate lim x→π/2 sin x / cos x.

Evaluate lim x→π/2 sin x / cos x.

• A. DNE (left: infinity, right: -infinity)
• B. 0
• C. ∞
• D. -∞

Answer: (A)

The limit tests how tan x behaves as x gets close to π/2. Since tan x = sin x / cos x, near π/2 the numerator stays near 1 while the denominator cos x goes to 0, so tan x grows without bound. But the direction depends on which side you approach from. From the left (x < π/2), cos x approaches 0 through positive values, so tan x goes to +∞. From the right (x > π/2), cos x approaches 0 through negative values, so tan x goes to -∞. Because the two one-sided limits are different, the two-sided limit does not exist. The graph of tan x has a vertical asymptote at π/2 with opposite infinite directions on either side.

The limit tests how tan x behaves as x gets close to π/2. Since tan x = sin x / cos x, near π/2 the numerator stays near 1 while the denominator cos x goes to 0, so tan x grows without bound. But the direction depends on which side you approach from.

From the left (x < π/2), cos x approaches 0 through positive values, so tan x goes to +∞. From the right (x > π/2), cos x approaches 0 through negative values, so tan x goes to -∞. Because the two one-sided limits are different, the two-sided limit does not exist. The graph of tan x has a vertical asymptote at π/2 with opposite infinite directions on either side.